AP Calculus BC Review: Unit 10 - Infinite Sequences and Series

Overview

• Final unit in AP Calculus BC series.
• Focus on Infinite Sequences and Series.
• Important for understanding calculus concepts and preparing for the AP exam.

Infinite Sequences and Series

• Infinite Sequence: An infinite set of terms.
• Infinite Series: A sum of infinite terms.
  • Example: 1 + 1/2 + 1/4 + 1/8 + ...
• Convergence and Divergence:
  • Convergent Series: Limit of partial sums is finite.
  • Divergent Series: Limit of partial sums is infinite.
  • Alternating Series: Series where terms alternate between positive and negative.

Tests for Convergence and Divergence

1. Nth Term Test (Divergence Test):

   • If limit of a_n as n approaches infinity ≠ 0, series diverges.

2. Geometric Series:

   • Converges if |r| < 1.

3. Alternating Series Test:

   • Converges if limit of absolute value of a_n as n approaches infinity = 0.

4. P-Series Test:

   • Converges if p > 1.

5. Direct Comparison Test:

   • Compare unknown series to a known series.
   • Converges if unknown series ≤ known convergent series.
   • Diverges if unknown series ≥ known divergent series.

6. Limit Comparison Test:

   • If limit of a_n/b_n is finite and not zero, series has the same convergence as known series.

7. Integral Test:

   • Continuous, positive, and decreasing series can be compared with improper integrals.

8. Ratio Test:

   • Converges if limit of a_{n+1}/a_n < 1._

Absolute Convergence and Conditional Convergence

• Absolute Convergence: If |a_n| converges, then a_n converges.
• Conditional Convergence: Alternating series converges but absolute series does not.

Power Series

• Power Series: Involves terms with x, e.g., (x-c)^n.
  • Example: (x-2)^n/n!
• Interval and Radius of Convergence:
  • Use Ratio Test to determine where the series converges.
• Taylor and Maclaurin Series:
  • Approximate functions using series expansions.
  • Taylor: Centered at x = a.
  • Maclaurin: Special case centered at x = 0.

Error Bounds

• Alternating Series Error Bound: Error is the absolute value of the next term.
• Lagrange Error Bound: For non-alternating series, involves maximum value of the (n+1)th derivative.

Practice Problems

• Practice using various tests to determine convergence or divergence.
• Derive Maclaurin series for functions such as 1/(1+x^2).
• Use error bounds to estimate the accuracy of series approximations.

Final Advice

• Good luck on the AP Calculus exam.
• Continue practicing problems and reviewing content.
• Refresh knowledge before exam day for best results.